What size wire and EMT conduit for 250 amps?
From NEC 2023 Table 310.16 at the 75 degree C column: copper needs 250 kcmil, rated exactly 255 amps. Aluminum or copper clad aluminum needs 350 kcmil, rated 250 amps. Both land right at or just above the 250 amp mark, so there is no room to downsize, and any long run or added derating pushes you to the next size up.
Conduit size for the run
For three 250 kcmil copper THHN conductors plus a neutral in EMT, Annex C Table C.1 allows up to five 250 kcmil THHN conductors in 2 1/2 in EMT, so 2 1/2 in handles three or four conductors with room to spare. If you are pulling 350 kcmil aluminum XHHW, four conductors need 3 in EMT. Add the equipment grounding conductor to the fill math when one is pulled: #4 copper per Table 250.122 for a 250 amp overcurrent device barely moves the numbers, but count it anyway.
Do not forget these
First, termination ratings: the 75 degree column only applies if breakers and lugs are rated 75 C, which is standard on modern service equipment. Second, voltage drop on long runs: at 250 amps even 250 kcmil copper drops about 3 percent at 150 ft on 240V, so feeders to detached buildings often go up a size. Third, if this is a dwelling service, Table 310.12 allows the 83 percent rule, meaning a 250 amp dwelling service can use conductors rated for 207.5 amps, which lets 4/0 copper or 300 kcmil aluminum work. That table is only for 120/240V single phase dwelling services and feeders that carry the entire load.
Grounding electrode conductor
With 250 kcmil copper service conductors, Table 250.66 calls for a #2 copper GEC. With 350 kcmil aluminum, it is also #2 copper. Size the neutral for the calculated unbalanced load, but never smaller than the GEC requirement per 250.24(C)(1).