I have an HP inkjet printer that I want to box up and store for a backup. The cartridges are about empty. What's the best way to store it so the ink doesn't dry up and make it unusable? I assume that it's best to remove the cartridges but how about the print head? Can it be cleaned or will it dry up and be ruined?

Is the print head not built into the HP cartridges? If the current cartridges are about empty, I’d scrap the current cartridges and buy fresh cartridges if/when I put it back in service.

One thing to keep in mind, if it is an older unit, and you keep it in storage for several years, it might be difficult to purchase OEM cartridges at that future time. You may have to go with after market or refilled cartridges.

its always good to have a new set of cartridges on hand...
I know nothing about print heads...

Do not set it down on a newly installed carpet to paint the window sill behind it...... Ask me how I know.

What’s the thing worth?

A $40 printer?

$100

$400 ?

I would remove and dispose of the nearly empty ink cartridges and if there is a separate print head I would flush it with alcohol to remove the residual ink.
You can find info on how to do the flush on YouTube

Depending on how long you plan on storing it, it may not be worth keeping around. I've had two printers I had to throw away because Windows stopped supporting them after a time and they wouldn't work with my computer ... this usually happens when Microsoft comes out with their latest version of the OS- usually they stop supporting most equipment that is more than two versions old IME... My Windows 10 won't support my Windows 7 compatible printer, etc....

Also, if you store your printer for a relatively long period, the ink at the injectors on the ink cartridge will dry up- blocking the ink injectors and usually contaminating the cartridge eventually. Best to remove the cartridges, clean the ink printer head on the machine, and buy new when you take it out of storage if it is still serviceable with your computer...

I have no idea how long I'll have it. There's no market at all for used printers, no matter how good they are. I'm getting a new one and just want to keep it just in case. Most likely, I'll not be using it again and since I can't sell it, I'll just stick it in a back room until I get tired of looking at it.
Since I posted, I've read that the best thing to do is run the clean cycle a couple of times and leave the old cartridges in it. Then use it at least once every 6 months or so. I suspect that the cartridges will dry up before long and I'm not going to spend $100 for a new set of them.

I'd throw it away or sell it for what it's worth which is pretty low. Print heads are notorious for drying up over time and you're screwed. You'll never get the thing to work again. I just threw away a Canon ink jet we had for a long time that did that.