It rather depends on what is real. 24V 6.5W is not 24V 0.5 A - but I think the latter number is just what the default (passive) POE power brick is, and they are oversized somewhat by default.
24V 0.5A would be 12W. 24 of those would be 288W and require the 500W switch.
6.5W X 24 ports would be 156W and the 250W would be fine.
Actual data update - not having the time free to muck up my network for this pro bono question, I checked the POE-Brick input against my Kill-A-Watt, and got a peak actual draw of (drumroll, please) 4 Watts (even with the radios blasting away on high power, which is not how I normally run them.)
VA is a bit worse at 8, but that's just the poor 120VAC power factor into a lightly-loaded 12 Watt AC/DC power supply. So, whatever inefficency is involved in the power brick and the draw of the radio is 4 watts or less (observed.) Call it 4.499 if you like, or call it 5 - regardless, this would appear to confirm the 6.5W worst case maximum draw as a very reasonable specification. To the original question, this means that 23 or 24 APs would be easily powered from a 250 watt (24V passive) POE switch.
If 802.3 af/at is something the OP is more comfortable with, the (slightly more expensive) UAP-AC-Pro model is of that 48V POE flavor, but it has a maximum draw spec of 9W, which likely means that a 250W POE switch will not be able to run 23-24 of them, since the Watt rating of POE switches seems to typically include the power needed to run the switch itself.