Lithium-Ion electric car batteries would do the job, but they are too expensive and I am not sure about their lifetime. A pack of ordinary lead acid car batteries seems to be a more affordable solution but I don't know if they are stable under high power demands.
You'll reget your choice if you select lead acid instead of lithium ion.
Any lead acid battery user tells you that you have to minimize your depth of discharge and immediately when partially discharging the battery, charge it as fast as you can. It's a race. Failing in that race, leaving the battery for hours (or worse, even days) partially discharged means the battery lifetime suffers. Complete discharges also let the battery lifetime suffer very fast. So with solar, if you have a sunny day and a cloudy day following the sunny day, being partially discharged during the cloudy day will kill your battery very fast. Lead acid is only realistic if every single day is sunny and you live in latitudes where solar power is not limited to summer season. Otherwise it'll fail very fast.
So lead acid batteries are batteries that really like to be fully charged. If not all the time fully charged, the lifetime suffers.
Lithium ion batteries in contrast degrade fast if fully charged. However, leaving few percent from the charge helps to remedy that situation. So for example 95% charged lithium ion batteries last quite long. Electric vehicles usually have a configurable charge limiter, allowing you to regularly charge to only 95% but then when you make a long trip, temporarily remove the charge limiter and charge to 100% prior to the long trip. Plug-in hybrid vehicles usually have a charge scheduler, meaning you tell that you depart at 8 o'clock and then the charger fully charges the battery to be just ready at 8 o'clock. Then it spends minimum amount of time in the fully charged states. Mobile phones also these days have a charge scheduler. If you set an alarm at 8 o'clock and leave the battery charging, it charges it just in time to be fully charged at 8 o'clock.
About the best use for lead acid batteries are computer UPSes. They have a battery to provide backup power when the grid power fails. A computer UPS battery is almost never discharged, because power failures occur usually more rarely than once a year. So a typical UPS lead-acid battery stays all the time fully charged. Even then, the users have to replace the battery usually once every 3 years.
If you use lead acid as a house battery, it will have a far worse life than a computer UPS battery. I'd say every 1-2 years you need to replace your batteries.
Also lead acid suffers from Peukert effect. The faster you discharge it, the less capacity they have. Lithium ion does not suffer from Peukert effect.
So, the only real solution is lithium ion. Todays there are other lithium ion batteries than the very dangerous lithium cobalt oxide batteries of old used in computer electronics. The newer chemistries such as lithium iron phosphate are reasonably safe and stable so they won't start a fire. 10 year lifetime is a realistic possibility if you don't charge to 100% but rather charge to 95%. You can get probably ten thousand cycles, perhaps more, if you charge to 95% and never discharge to 0%. Lithium ion batteries are used in power tools so faster than 1 hour discharge is a possibility. So 5 kWh battery would allow 5 kW discharge. There is no "memory effect" so you can charge a partially discharged battery (actually with nickel cadmium batteries the "memory effect" isn't ever encountered in terrestrial uses, it only limits battery use in satellites where the battery is all the time discharged to the same power level, usually instead of "memory effect" the cause for battery degradation is bad chargers which torture the battery by overcharging).
About price: they are starting to approach 100 USD / kWh. So 500 USD (without taxes, without charger, without inverter) could be a realistic possibility. However, expect to spend at least several thousand USD if you want charger and inverter along with the battery.