A lot of good answers here. Every electrical install should be considered on its' own merits, not necessarily what the "code" calls for. Actually the wire size tables in the NEC are pretty good if you take ALL the factors into consideration. I'm way out of date, well, my knowledge is, so I stand on what I remember rather than the current code. The best call I can make here is what I've done, not what a code calls for.
The first consideration is distance. Most codes call for wire size increase for each hundred feet. The next is for anticipated load. If you want a fan and light fixture now, what about in the future? The next consideration is insulation class, you don't want romex cable for open air, exposed install. If you will be burying, use UF(direct burial) or NM conduit. If aerial, use SE Triplex. If along a wall, use SER, but that isn't likely. If you opt for conduit, install SE cable.
For a 30 Amp feeder, I would recommend a minimum of Nr 6 AWG SE aluminium for the distance involved. Since you are running such a large wire, go the extra mile and make it a 220 Volt service. It won't cost that much more, might even be cheaper because of availability. And you will have it in place if(when) you ever need it. Even as simple as a second 120 volt circuit
My "barn" is actually a heavy shop building for welding, blacksmithing(?) and lumber(actually anything I don't need right away) storage. It is only a hundred feet or so from my residental service. I have a feeder of Nr 2 SER copper to the inside shop(~50ft) with a 75 Amp breaker and a feeder of Nr4 Triplex aluminium submain aerial to the barn. The barn load is nominally a few lights and a dusk to dawn flood.
But, and this is what matters, there is a buzzbox welder and a travel trailer feed, and a generator backfeed to the residence for outages. So, a 50 Amp circuit, minimum. I used what I had on hand so it may be a little large. But oversize is handy if I want to add something in the future.
Guesstimating an install remotely is difficult at best. Call it the largest wire size needed. Plus two sizes for distance. Plus another wire size if you opt for aluminium. It's a helluva lot cheaper than copper, but there is a small price to pay. You got to allow one wire size just for material. For long spans(>100ft) I usually recommend it. You're pushing the limits for a DIY project. Make sure that everything is large enough wire. And suitable sized fusing. Don't trust the NEC tables for this without allowing for distance.
.