All the HP/Agilent's I know, particularly the HP branded ones, have a power on self-test function POST, or a full separate self test. This is one good reason HPs are good on the second hand/eBay market, because there is a simple way to find out all is well, for the most part. Depending on age of your HP device, a POST can test 75-100% of a device's functionality.
For the 34401a, on power on, it does an essential self test, not the complete one, which takes 15-30 seconds more.
To perform the complete front-panel self-test is as follows:
Hold down Shift as you press the Power switch to turn on the
multimeter; hold down the Shift for more than 5 seconds. The selftest
will begin when you release the key.
It will then report full status.
This meter is an industry "standard", and it has a lot of 3rd party software and accessories. But it is obsoleted by its newer models, all are still sold by Agilent. I think its the 34410a and 34411a, they are priced identical if bought new: you get more bandwidth, functions, and higher sampling speed in the new models, but the 01a is well entrenched since the new models haven't been battled tested in real world use as much as the 01a.
Is it worth $350, yes. That's the going rate for a fully functional used one, I've been looking for one myself.
Its main competitor is the older bench model HP 3456a, which has a slight edge in absolute accuracy in Vdc, however it has no amp function which the 01a has. It typically goes between $50-100, if working status is unknown, and $100-300, if working. I bought 3 3456a, and planning to buy a few more. The reason for having multiple is to calibrate them against each other, use one as transfer reference, and backup each other since they are 20-30 years old, and I can find them dirt cheap over the 01a. Lastly, older bench DMM are easiler to self service in terms of parts, whereas the 01a has many custom ICs that can only be repaired by buying another 01a and strip it for parts.