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For a high-end gaming rig at least 1000 VA, but that will still only have about 120Wh of energy max.

Remember that the VA rating is also used for calculating the time of running on batteries 5 minutes is the normal time you can have the VA rate prior to running out of energy.

I do prefer the 2200VA class as they actually are better designs and have higher battery capacity and are often expandable.

To answer that question, we would need to know how many watts the PC consumes and how long it will be needed to run without AC power.

A UPS should have an optimal KW/hour rating. If your PC consumes 500 watts, or .5KW. and you need it to operate for 30 minutes to withstand a power outage, the calculation we used is to specify 2x the capacity or in this case 1KW/hour of battery backup. We did this because batteries deteriorate over time, and it’s common for power to “flutter” a couple times before remaining stable.

The UPS are rated in KVA (not in Watts) at normally 0.8 pf lag. It does not mean a 1 KVA inverter (rated at 0.8 pf lag) can support 800 Watts load. It supports about 700 Watts only. The anomaly is because of the capacitive filter of inverter which draws leading current from inverter which cancels part of lagging load current. This feature favours inductive loads. Resistive loads do not compensate for filter current hence the lower rating on resistive loads. If the load is capacitive, situation is worse as both load & filter currents directly add in inverter. The 1 KVA 0.8 lag pf rated inverter can supply only 600 VA at 0.8 lead pf loads.

Your load is 650 Watts PSU which will be slightly capacitive nature because of the filter in the PSU. Going by the above discussion you need higher than 1 KVA (rated at 0.8 pf lag) UPS. Better go for 1.5 KVA UPS (considering non-advisability of fully loading the inverter).

The other answers pointed out aspects of battery sizing. Pleas note the battery sizing depends on actual watts load (not on VA load) & the time of backup required. Batteries are rated in nominal AH they can steadily deliver in 10 hours (vented Lead-acid cells). But they deliver only 50% of nominal AH if asked to deliver quickly in 1 hour. Also battery capacity degrades with time (say) by 30% by end of their life (4 to 5 years for lead-acid cells). Hence if you require 1-hour backup until the end of life of the battery:

If you are looking for 1 hour backup time for your 650 watts load with an inverter efficiency of 80%, you need a battery rating of 650 W / Inverter efficiency 0.8 / 1-hour quick discharge factor 0.5 / life factor 0.7 = 2322 watts battery. At 24 V battery volts, it translated to approx. 100 AH nominal rating (10-hour rating) battery.

You need 2 nos. 12 V 100 AH sealed lead-acid batteries to give a minimum 1 hour back up for the UPS.

If you have additional loads (other than 650 W) like monitors, routers, etc, the battery size increases proportionately