Upgrade the PSU. Even if you had a 'magical' PSU which could put out the necessary power in spite of its rating, you still wouldn't have the required connections for the PSU. If I remember correctly, each 7790 should sport a single six pin aux power connector because it draws over 85 watts each at full load (170 watts total for two cards). Adding in just rough numbers for the rest of your system and you'll easily more than top 240 watts for your two cards, basic system, hard drive, and CPU.
How much PSU will you need? Well, step back before you buy the cards and make sure of a few things:
1) that you have two PCI-e x16 (full length) slots available. Obviously if you have only one full length x16 PCIe slot twin cards do no good for you.
2) determine if your chipset will support SLI or Crossfire. This is rather important because even if it is a full length slot, it might only be -wired- for x4. What you need is a minimum of 1 @ x16, and 1 @ x8. That's all determined by the motherboard chipset which provides the PCIe lanes or the CPU (which you didn't specify).
So, if you have two PCIe full length slots AND they are wired in a minimum x16/x8 configuration THEN you determine the PSU.
As mentioned, the cards together will draw 170 watts at load, motherboard about 50 watts, CPU is a guess - if we guess 90 watts that would run a pretty wide gamut of choices from AMD and Intel, toss in about 20 watts for a drive or two and a fan and it guesstimates to about 330 watts. That would be everything running full tilt and worst case scenario (from the PSU's perspective).
Now, PSU's run in their 'sweet spot' in the 50-70% load range, so divide 350 by .70 and you're looking at needing a minimum of a 500 watt PSU. This would give you some headroom (PSU's don't like to run 100% for long periods of time) and when you buy the PSU make sure it has two 6 pin PCIe power plugs (adapters are not an ideal solution).
As to your question as to whether your current setup can run reliably on a 250w PSU with a graphics card (as you bought it), the answer is - probably. A lot depends on exactly what you've got which (again) you didn't really specify. A light weight video card (40w), CPU (50w), mobo (50w), drive + fans (20w) would fit easily under the 240w max your PSU can supply.
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