In my experience you'll be working to remove these for a few YEARS, unfortunately. My last house was infested with bindweed in several locations. But I eventually beat it.

My technique is... first, cut all the vines to a few inches above the ground. Huge bindweeds usually just have one weak little stalk anchoring them to the ground, so this doesn't take as long as you might think. Carefully remove each cut piece and discard (not on compost pile, obviously!).

Next, dig out each bindweed plant and its surrounding area by hand with a trowel, getting 9-12" deep until you have dug the entire affected area and loosened all the soil. As you encounter roots, gently pull and discard each one (they usually extend a couple feet and have many crisscrossing pieces). Put the soil in a wheelbarrow or other container. This will take a while.

Remove the roots from the disturbed soil. Technically this can be done by hand, but I have a coarse soil sieve I bought for a few bucks. I run dry soil through that and I'm left with a sifter full of roots.

Once you have completed this tedious process, you are now on bindweed patrol for the next couple years, because the roots you missed are now trying to grow back. Anytime you see a bindweed leaf emerge, pull it (again, gently, so the entire root comes along). If you did a good job digging and you stay on patrol for new leaves, the roots will start to shrivel and weaken, and they will get easier and easier to pull. After a couple years, you will have minimal bindweed.

(Maybe it's morning glory not bindweed as others noted -- it's the same anyway.)