Can I use Accurate to ready a unit that is always on the playing field? Say I moved it from a battlefield to base. Can I pay the Accelerate cost?

No. Accelerate only works "as you play" a unit — it's an optional additional cost paid during the steps of playing that unit as a card (Rule 805.1.a, 805.2). A unit already on the board cannot pay Accelerate; it has no function there (Rule 805.2.a, 805.3).

Moving a unit from a battlefield to your base (Standard Move or an effect) is not "playing" it, so you can't pay the cost, and the move doesn't ready it — the unit remains in whatever state it was in unless the effect specifically readies it. Accelerate only ever applies when the unit enters play from hand or another non-board zone.