1970 Monte Carlo engine issues

Recently I bought a 1970 Monte Carlo with the original engine and transmission but they have been rebuilt by Jasper some 60,000 miles ago. The engine is a 350. So I can’t get it running right, or for very long. It will take a few minutes to start. If for some lucky reason it does start, then it will idle rough, it will get bogged down when you accelerate, and it always dies with a backfire. Sometimes it backfires multiple times before it dies. I have rebuilt the carburetor, replaced the plugs and wires, and it has clean gas in it. What can I do to fix it?

Clarify “backfire”. Does this event happen out the exhaust pipe, or back through the intake/carb?
Loosen the distributor. Turn it a 1/4 inch to the left, and run the engine. If worse, or no better, turn the distributor a 1/4 inch to the right, and run the engine.

the backfire occurs through the carburetor. I also failed to mention that it has an HEI distributer. Could the problem be in the Ignition coil or the pickup coil?

Check the timing. If timing checks out o.k. after re-setting it, time to check the timing chain.

So this is the engine that has been sitting for years in this post: http://community.cartalk.com/posts/list/845812.page ?

Did it run well before it sat for years? Explain “It will take a few minutes to start.” How did it run before you rebuilt the carburetor, and EXACTLY what was involved in that rebuild, because I’d bet that you still have plenty of carburetor problems, beginning with it being so gummed up from varnished gasoline that nothing in it works right.

Also, double check to make sure you don’t have two (or more) of the plug wires incorrect.

Define “rebuild the carburetor”. I don’t know how this was performed but a Quadrajet can be a bit touchy to rebuild and they do have a few quirks.
A proper rebuild means soaking the disassembled carb in carburetor solvent for at least an hour followed by thoroughly rinsing it out with water, and then followed by blowing every single passage out with compressed air to verify that no passages are obstructed.

New float, jet well plugs, inspection and replacement of the choke pull-off, etc. should all be part of the rebuild. There are quirks that can prevent the secondary butterflys from opening and this can also contribute to a problems similar to yours. It does sound like a carb problem to me.
Assuming the ignition timing is correct of course, the advance weights are not hanging, etc.

First year of the Monte Carlo and it’s a nice car anyway! I used to have a '72 w/350/4BBL and it was great vehicle. Would get 19-20, and sometimes 21, MPG on the highway with the A/C running. Not shabby for a non-computer controlled, carbureted vehicle with 3:08 gears.

The choke may not be closing all the way. If it is closing, maybe the pull off isn’t working. The timing may be retarded, the vacuum advance may be broken or frozen, The choke stove and or pipe may be missing, the float level may be too low. The wires on the ignition pickup may be frayed, the module may be toasted. The center electrode may be missing in the distributor cap, the top contact of the rotor may be twisted the outer contact could be broken. You name it. Time to start checking. Ignition wires may be shot. Coil cracked.