I would like to know if there is a solid state device I can use to back up my digital files without a computer when I am traveling.
It seems like it's harder to find photo backup devices these days than it used to be, perhaps because large memory cards are relatively cheap and pro-level cameras can often use two cards at a time, automatically creating a very small, very portable backup. In any case, there are still some devices available; to find them, search for memory card backup device. Devices range in capability from something that's barely more than an external hard drive to an entire system that could probably serve the needs of the Associated Press. One of the more promising ones I found (and I'm just browsing -- I have no direct experience with this thing) is the Sanho Hyperdrive, which is essentially a fancy hard drive case with a screen and slots for reading several types of memory cards. You can get it in a range of capacities from 0TB (empty case, you supply the drive, $250) up to 2TB (hard disk, $700). It looks like this:
I know you said you were looking for something solid state, so you might consider buying the empty version of this device and inserting a SSD (solid state drive). The B&H site says that the HyperDrive supports SSD's up to 1TB. I recently bought a 750GB Sanyo SSD, and it cost about $180.
So, that's one option. Another one that I found interesting is Unlimited SD Card Backup in the Field, an Instructables article in which the author describes a simple-sounding procedure to cut a hole in the side of a Western Digital My Passport Wireless hard drive to make it hot-swappable. The drive itself is interesting: it's a battery-powered hard drive with a built-in SD card reader, and it can be set up to automatically back up any inserted SD card. The author's idea is to cut a hole in the case so that you can easily replace the drive without disassembling the device. Since you're looking for a solid state solution, you'd pull out the original drive and insert a SSD.