Using the Southwire Voltage Drop Calculator and 350 feet, 1/0 aluminum wire, 240V and 70A, the result is 4.25% drop. That's not too bad. However, at 120V it is 8.50%. So a key issue is 120V vs. 240V loads and how well balanced the system is.
I would definitely consider upsizing. At 2/0, the 240V 70A drop goes down to 3.32%, which is close to the mythical ideal of 3%.
You may also want to consider conduit instead of cable. Yes, conduit + wires will probably cost more in terms of material than direct burial cable. But you may make up for it in labor/installation cost, and you then have more/better options for dealing with future upgrades or (if something messes up wire in conduit, which is less likely than something messing up direct burial cable but could still happen) repairs. Plus you may be able to run a fiber optic cable for internet access through the same conduit.
Burial depths:
- Direct burial: 24"
- PVC Conduit 18"
- Rigid Metal Conduit or Intermediate Metal Conduit 6"
Note that conduit depth is 18" under certain types of driveways and parking areas and 24" under others. But those are usually only a small part of a 350' run.
350' is a long distance. If it is basically flat, clear area then 24" isn't so bad because you rent equipment and/or hire a company to trench through the whole thing quickly. But if it is "over the river and through the woods" then trenching deep is a lot harder to do.