Is it at all possible to have an external "drive" offering a PCMCIA slot, connected to my laptop by USB?
No. It is flatly not possible.
A PCMCIA slot - rather a CardBus slot in more recent laptops, although CardBus slots will take PCMCIA cards - is basically a PCI slot in compact, hot-pluggable form. You can have a USB controller that's a PCMCIA device (i.e. plugs into a PCMCIA slot), but you can't implement a PCMCIA slot with a USB device. (PCMCIA is older, and it's actually ISA-like rather than PCI, but the same principles apply.)
A device plugged into a CardBus slot has direct access (via DMA) to the host's RAM. And CardBus devices can be directly addressed as "memory" in the address space of the host CPU. (This is not at all the same thing as a "USB memory" stick; or an SD card, or the like; the "memory" in such devices does not appear in the host CPU's physical address space.)
These capabilities cannot be implemented in a USB device (i.e. a device plugged into a USB port). USB doesn't have any way for devices to do those things. Those concepts just don't exist on USB. There's not even any way in the USB protocol to ask for them. It would be like trying to use a can opener to wash the dog.
As you've noted, these slots are now obsolete, along with PCI. With PCI Express (PCIe) we have a new external slot called ExpressCard. What you need to do is find an ExpressCard equivalent for the PCMCIA (or maybe CardBus) card you used to use, and buy a laptop with an ExpressCard slot.
Either that, or buy old laptops on eBay or similar and keep using your existing cards.
Best Answer
If you do not need to move around much you can combine an express card to PCI-e adapter and combine that with a common PCI express to PCMCIA adapter.
The nice thing about this is that it is a very flexible solution, allowing you yo use common PCI-e cards (including PCI-e to PCMCIA, or even PCI-e graphics cards).
The downside it that it is not very portable. If you only need the PCMCIA card at one location you could use this and just unplug the card. If you need it in the field it becomes unwieldy.
Note that technically this is relative trivial since express card basically is a single PCI-e lane and a single USB connection. However you will need to find the right drivers for your PCMCIA card. That might be a challenge on modern OS's (PCMCIA is based on old ISA. That thing from the IBM XT time).