Taking into consideration the landmass of the whole of the United Kingdom, areas of open country where windfarms could be placed, areas of country designated as wildlife havens and also areas of particular scenic beauty or with notable wildlife such as raptors where windfarms or renewable energy acquisition sources should not be placed. And taking into consideration the population of the whole of the UK, the number of households and businesses and the average electricity consumption of each, could the UK presently sustain itself entirely from renewable energy sources (such as wind and solar) and not coal, oil and gas? Also I consider nuclear sources as non-renewable as they do produce harmful waste and do consume fuel albeit in a much more efficient manner than fossil fuels.
Summary: yes, the UK can be powered practically by 100% renewables
Technically it's fairly straightforward. Subject to some fairly reasonable assumptions, and the commercialisation of technology that's already proven in the labs, it's also economically efficient too.
Total UK energy demand is about 200 GW, of which 40 GW is presently served by electricity. (source: DUKES 2014)
We don't know how efficient wind, PV and biomass can get, in terms of yield per unit surface area, as it's never been an issue: surface availability is not a constraint: cost and permissions are the constraints. So if we take existing yields per unit surface area, we are erring on the safe side.
Offshore wind yields are of the order of 2 or 3 Watts per square metre, averaged out over time: that's equivalent to 2-3 GW per thousand km2. That's not installed nameplate capacity, that average delivered power.
Getting offshore wind equivalent to that 40 GW of current electricity demand would require 13 to 20 thousand km2 of surface area. And for 200 GW, that would require 67 to 100 thousand km2 of surface area.
The surface area of the UK's seas, the Exclusive Economic Zone or EEZ, is about 770 thousand km2, so at the very most we're talking about less than 15% of the EEZ, even if we had no land-based renewables at all, no energy efficiency improvements, and no increase in energy yields per unit area. So that's not 14-15% of the EEZ along with land-use for onshore wind and PV - that's instead of those. Now, no one's suggesting using offshore wind only: the calculation is just to give an indication on the outer limit, and to show that the available resource is not at all a constraint on going 100% renewables.
Now onshore wind and PV are both cheaper than offshore wind, so there are good reasons to use those as well as offshore wind; the onshore resoure is smaller - various estimates put it at between 20GW and 50GW average output, at about 2GW per thousand km2. Hence 20GW would be 10 thousand km2, and 50GW would be 25 thousand km2. The UK's surface area is just over 240 thousand km2, so we're talking about 4-10% of UK land area. Easy enough to avoid Areas of Outstanding National Beauty, and Sites of Special Scientific Interest.
The photovoltaics resource is huge, even in cloudy old Britain. The limiting factor on PV isn't available area: it's that its generation is concentrated in spring to autumn, with very low supply in winter, when energy demand is highest. Integrating PV is pretty easy up to the point where capacity is equal to average demand; actual average output would be about one-tenth of that. So, integrating 4-20 GW of PV output is fairly straightforward (taking demand to be in the range of 40 GW of present electricity demand to 200 GW of present all-energy demand). PV generates at about 5W per m2, equivalent to 5GW per thousand km2. And so we're talking about 1-4 thousand km2, i.e less than 2% of UK land area, and absolutely no need to put PV on land currently used for growing food crops.
balancing demand and supply, and electricity versus other vectors
So, there's really no shortage of energy there for the UK to harness: there's way way more potential there than we could possibly use. And protected areas can stay protected.
A sufficient quantity of resource on average is only part of the story. There are two other parts. One is balancing demand and supply; the other is the form of energy. Presently, only a fifth of energy demand is met by electricity. Gas, coal and oil products meet the rest. In some cases, electrification is pretty straightforward - for example, for heating, and urban car use. And electrifying car use would have the additional bonus of reducing total energy demand by about 25 GW, as electric cars are about four times as efficient as fossil cars. Electric heating is also typically more efficient than gas heating, though not usually by such a large margin.
There are some uses where getting off fossil fuels looks a lot more challenging: in particular, air- and water- transport. There are a couple of options: biomass, and synthetic fuels. Both work in the lab and in prototypes, but given the world is awash with cheap oil and no sane carbon price, getting either of them commercialised is economically challenging at the moment.
How we electrify these things matters a lot, because it influences how we match demand and supply. Electrifying heat is great, because heat storage is very cheap, and hugely scalable, offering balancing services at the scale of milliseconds to months. As would synthesising fuels using electricity. Electrifying cars offers balancing services at the scale of milliseconds to a day or two.
That leaves some industrial processes such as steel-making that currently rely on coke. Now, coke isn't the only way to make steel - what you need is lots of heat, and some carbon. Coke is one way to do that, but it's not the only way - there are lots of sources of carbon other than fossil fuels, and lots of other sources of heat too. So, technical solutions are there, but again the economics are not currently favourable to commercialising these solutions.
efficiency of space use
A few years ago, some people started worrying about whether this space use was efficient or not. That turned out to be irrelevant, for a couple of reasons. Firstly, when we do the maths, we find we'll only talking about a few percent of land and seas at most. And secondly, and more interestingly, adding PV or wind to land or seas makes those areas more efficient, not less. For example, take a sheep farm. It's currently used for farming sheep. You can add both PV and turbines to it, and now the land is being used much more efficiently: you get electricity when it's sunny; you get electricity when it's windy; and you still get sheep all year round.
Renewables repeatedly come out top in polls of public acceptability: they receive far higher levels of approval than any other generation source. Electric vehicles offer a large number of advantages over fossil-fuelled vehicles, and as costs come down, uptake increases.
an example solution
We could get about 6 GW from tidal barrages, 10 GW from PV, 20 GW from onshore wind, 114 GW from offshore wind, and 50 GW reduction in demand from energy efficiency measures that improve the quality of people's lives, such as higher-quality lighting, and well-insulated buildings. That would mean the renewables would enhance the efficiency of use of about 5% of our land area (though the physical footprint would be less than one-tenth of that: wind turbines occupy very little land), and about 7% of our seas. Now, there's also wave, tidal stream and solar thermal too, so I've only looked at part of the picture; but enough of the picture to be able to answer the question in the affirmative:
yes, the UK can be powered practically by 100% renewables.
Energynumbers gives an excellent summary of the purely technical aspects. While I might quibble with a few specifics, I'll accept that - subject to developments of storage technologies and some substantial demand reductions - 100% renewables in the UK is technically possible.
However, technical considerations are not the only ones. There are two other aspects that spring to mind that make this ambition substantially more difficult to achieve.
Firstly, public acceptance: Will the British people put up with (to use Energynumbers' figures) 4-10% of UK land area hosting wind turbines along with 15% of the EEZ seas, 2% of UK land area hosting PV? How about the environmental impacts of tidal barrages, and then the things that actually affect people in their homes - further energy efficiency measures, and "forced" demand-side management (either statuatory or with very strong price incentives), etc? Remember the tabloid horror about "Banning the light bulb"; we've already seen the Daily Mail's first response to modern demand management, which read "GOVERNMENT WANTS TO TURN YOUR FRIDGE OFF". Ultimately, the UK can only go fully renewable if its people are prepared to accept that they can see the infrastructure.
Secondly, how do we get there from here? To take an example, the scenario described involves a nationwide takeup of electric vehicles, and a portion of connected vehicles' batteries being used as distributed storage. That's a great end-point to arrive at. But first of all, we need the mass takeup of electric vehicles. But that isn't feasible without major grid reinforcements and upgrades, and increases in generating capacity. But there's no commercial incentive to do that until the uptake of EVs is clear; indeed, current OFGEM rules would probably not allow the grid operators to do the upgrades based on a hypothetical demand, because that would represent "poor value for the consumer". Catch 22. To make all of this work, a substantial amount of risk and short-term financial loss needs to be taken on to get from here to there, and I suspect that that that is ultimately a political question: Is there any likely UK government in the near future that will be prepared to pledge tens, maybe hundreds of billions of pounds of public funds on a grand project that will not come to fruition while they are still in power? Maybe. High speed rail is a precedent there. But the political setting will need to be just right.
 Note how for years people have said "We don't want wind turbines here, put them offshore", but now that large offshore farms are planned, there are objections to these too on the basis that they will "dominate the horizon" when viewed from the coast.