While Amazon offers a battery-powered version of the Echo in the form of the Echo Tap, it doesn't have the always-listening capabilities that makes the original Echo and Echo Dot so great. However, you can run the Echo Dot on battery power and have it last for days--all you need is a USB battery pack.
Update: Amazon now offers always-listening on the Amazon Tap, but if you already have an Echo Dot, this is still a worthwhile project!
The Amazon Echo is made to be a stationary device--it's meant to be set up somewhere in your house and never moved around after that point. However, it could be game night and you want the Echo close by, or you want the Echo somewhere where an outlet just isn't nearby. Heck, maybe you even want to take it camping over the weekend.
There are already products on the market that are specifically geared toward cutting the ties between your Echo and the outlet, but they're pretty expensive, and don't really provide you with the best battery life. Nonetheless, these products are perhaps the best option you have if you want to make your original Echo battery powered, as it uses a dedicated power cord, rather than a simple microUSB connection like the Echo Dot.
However, if you have an Echo Dot, there are much cheaper options that can provide you with a longer battery life and more versatility.
Get a Portable Battery Pack
You're probably familiar with portable battery packs, which allow you to recharge your phone on the go. Since the Echo Dot uses a microUSB port for getting power, you can use any ol' microUSB cable to power it. Then, to run off of battery power, simply plug it into any portable battery pack. The Echo will automatically boot up and be ready for use.
As far as what battery pack to get, I'm a big fan of Anker, and they make a beefy 20,000mAh battery pack that's available for $39. According to my own tests, the Echo Dot consumes around 570mAh of power every hour, which means a 20,000mAh battery pack could provide power to an Echo Dot for 35 hours straight, or almost a day and a half.
Of course, if you unplug it at night when you're sleeping and not needing to use the Echo Dot, you can make it last even longer, perhaps as long as three days if you really stretch it out.
However, this doesn't necessarily make the whole setup portable by any means, as in it's not really something that you can easily just pick it up with one hand and move. A 20,000mAh isn't exactly small, and they're pretty heavy for they're size. Plus, the microUSB cable connecting the Echo Dot to the battery pack can be a bit cumbersome, which is why you might be better off with a short cable that doesn't get in the way or tangle up.
Still though, this kind of setup at least frees your Echo Dot from the confines of the wall, and you can set it pretty much anywhere, no matter if there's an outlet nearby or not.
Bonus Upgrade: Get a Bluetooth Speaker/Battery Pack Combo
If you want to take things to the next level and even turn your Echo Dot into a full-fledged Echo of sorts, you can get a Bluetooth speaker that also doubles as a battery pack.
One Reddit user combined his Echo Dot and JBL Charge speaker, connecting the speaker to the Echo Dot over Bluetooth and powering the Dot via the USB connection from the speaker. The speaker only comes with a 6,000mAh battery, so it certainly won't last as long as the previous method, but it will give you at least a few hours of use depending on how much you're using the Echo Dot and the speaker.