Are there any types of DC motors that self-lock themselves?

A common way of doing this is with a self-locking worm drive The worm drive has a gear ratio that provides high mechanical advantage and depending on the helix angle of the gear, the output can't backdrive the input, so it is self locking. You can find one here at Servo city or try to DIY with hardware store threaded rod and an off the shelf gear.

Another suggestion is a ratchet and pawl mechanism that may be easier to DIY than a worm drive and much cheaper than purchasing it. Note that the rachet can only be driven in one direction.

There are other ideas I can think of such as using a solenoid to lock/unlock a catch that holds the belt in place when the motor is stopped, but they are more complex than what I describe above.

Please post back here if you do build one of these. I love mechanisms but find that few people are interested in building them.

You want a DC motor with a mechanical brake.

You can electrically brake (permanent magnet) motors by shorting the contacts together, but the braking force of this isn't very strong for small motors.

For your cord application, I'd look into some sort of clamp applied to the cord itself with a servo. Especially if it's safety-critical; I can't tell from the description whether this is the sort of cord used for opening curtains or the sort that might have heavy objects or people hanging from it.