Wave goodbye to "Back to My Mac." As of macOS Mojave, Apple is officially removing the helpful file and screen-sharing utility baked directly into the operating system. And while Apple has a few suggestions about what you can now use instead of Back to my Mac, they're less practical, especially since one "solution" is going to set you back a lot of money. Pfft.
If you've never used Back to my Mac before, it lets you accomplish two things: copy files from a remote system (that you've also set up with your Apple ID and Back to my Mac) and control that system remotely. Thankfully, that's not very hard to do with a number of other apps, especially those that are just as useful (if not more so) than what you'll find in Back to my Mac.
Controlling a Mac system remotely
I'll start with the second feature first, and you'll see why in a bit. Generally, I'm a big fan of Google's Remote Desktop app for Chrome. Even if you're an Apple purist and can't stand the thought of anything Google on your system — and you must be bitter if you have to use the iCloud web interface all day long — Google's app makes it incredibly easy to access a system from afar.
I use it to control my Windows desktop from my Mac laptop whenever I'm away from home (or am too lazy to get up and fuss with some setting).
The biggest perk of Chrome Remote Desktop is that it's incredibly easy to set up for even computer novices. This makes it even more ideal if you're the tech support person for your family or circle of friends. Rather than trying to explain a technological process in painstaking detail, you can just walk the person through setting up Chrome Remote Desktop.
They can then use its Remote Assistance feature to generate an access code, which you'll type in on your version of Chrome Remote Desktop to control their system from afar. You'll probably be able to fix their issue a lot faster than if you were trying to explain it to them, but you can always explain as you go to at least help them understand what you're up to.
That way, you can have the best of both worlds: a little energy-savings when you aren't at home, but the ability to dial in remotely without having to keep your PC up and running all the time. TeamViewer also makes it incredibly easy to share files between the two connected systems, which could also solve that missing aspect of Back to my Mac.
Accessing your Mac's files from afar
The easiest way to access your files remotely is to enable iCloud drive for your account on all of your systems (via System Preferences > iCloud). Do that, and any files you drop onto your desktop or into your documents folder will automatically synchronise to your other systems the next time you power them on.
You even get (a whopping) five free gigabytes to play with in iCloud. Go crazy.
There are plenty of other cloud services that let you do the same thing, and we've already covered most of them (and their costs). The only drawback is that if you need a specific file that isn't in one of your synchronised folders, you'll have to remotely connect to your system using one of the aforementioned apps and drag it in, then wait for it to sync to your remote system.
You can cut out the middleman if you get a little crazier and set up your own VPN server at home — typically an option on higher-end routers or network-attached storage boxes, or as a separate accessory to your existing network infrastructure.
Dial it up with an app like Tunnelblick, and your remote system will be able to see all of your other systems in your home network. You can then access drives you've previously shared and grab your files that way.
That all said, maybe the best option is to simply live in the cloud as much as you can. While you might feel a little "eh" about giving up your files to a third-party service (especially if your company forbids you from doing that with work documents, for example), it's a lot easier to access your stuff when you're not the one responsible for its well-being.