You could use a 13th Age style Engaged (base-to-base) / Close / Far Away abstraction, but all powers would need to be rewritten to support that.
You could use a Fate style Zones abstraction, but all powers would need to be rewritten to support that.
In a couple of the LFR adventures, there is an abstraction where, to see how many endless mooks you affect with a power (think horde of zombies), you take the size of the burst/blast plus 1. So an Area Burst 1 would affect 2, a Close Blast 3 would affect 4. Obviously this is highly abstract and can really punish players who were otherwise great at positioning. (I should easily be able to hit 4 targets with an AB1, not 2; etc.)
But anyway. Without significant rewrites, 4e needs the grid.
As for minis: I used and still use the flat cardboard “pogs” sold by Fiery Dragon Games and WOTC themselves (Monster Vault, Madness at Gardmore Abbey). Super affordable, durable, differentiated enough to tell the monsters apart.
PCs are actual minis, but you only need, ideally, 1 per player over the course of a campaign.