There's a good chance you might able to revive it yourself, even with limited/no technical knowledge. The battery in A3000s is known to leak but my (admittedly limited) experience tells me the damage caused is rarely extensive - i.e. not as bad as in A4000s - and usually not bad enough to prevent a machine from booting altogether. This probably comes down to a combination of less destructive acid on the battery itself (as opposed to the A4000) and the battery's less critical position in the motherboard. Like I said though, this only my experience (coupled with a good dose of speculation) and cannot guarantee that it holds more generally.
I think the advice given above -to remove the leaky battery with pliers *carefully* and gently wash off all the slime - is good would need to be done anyway.
For all you know the battery leak though may be unrelated. Here's a few other things you could try that require minimal intervention:
(1) Push down hard (but not 'too' hard) on all socketed chips. They have a tendency to become loose, especially if you've moved the computer about. Check that the daughtercard hasn't become unseated.
(2) remove/disconnect all peripherals - including zorro cards, accelerators etc. If that fails too, trying disconnecting the harddrive (cable and power leads).
(3) clean dust, spider cobwebs etc (not likely to be causing a problem but won't hurt) with a small brush and/or a can of compressed air.
(4) Confirm you have the amiga connected to an appropriate monitor (if using VGA, make sure the switch on the back in on the 30khz position)
If all the above fails and you have a second A3000 handy (or a friend with one) you could try swapping power supplies - another common source of failure.
I don't know of an amiga repair centre/guy in the US but in Europe there's a guy in Paris with legendary reputation. The A3000 weighs a ton though and posting it to Europe might be dear. If it turns out to be the motherboard and you're not too scared to disassemble it, shipping just that to him would be an option. I don't remember his name/alias but I'm sure someone here will fill me in.
good luck.
