The average PC user will probably want to stick with a cloud storage service instead of a remote file access solution. Place your files into a service like Dropbox, OneDrive, or Google Drive and they'll sync online. You can then access them from anywhere via your web browser, the service's mobile app, or its syncing client.
The upside is that your remote PC doesn't have to be powered on. The downside is that you're limited by the amount of storage the service provides. If you're worried about storing sensitive files online, you can always encrypt files before storing them in the cloud.
Using documents in the cloud
Microsoft Office users can save documents to Microsoft's free OneDrive cloud storage service, then access the Office Online website in a web browser to view and even edit the documents from wherever you are. Office Online is completely free. OneDrive is integrated into Windows 8.1 and is available as a free download for previous versions of Windows.
The Office Mobile applications for iPhone, Android, and Windows Phone also allow you to view and edit Office documents stored in your OneDrive account for free. Office for iPad provides a more powerful editing experience on the iPad, but note that all the mobile apps require an Office 365 subscription for editing (viewing is free).
Microsoft Office isn't the only game in town. Google Docs is completely free, runs in a web browser on your PC, and allows you to edit your Google Docs or Microsoft Office documents from anywhere via a web browser or mobile apps.
Wake your computer remotely
PCs have long supported a feature known as Wake-on-LAN. When Wake-on-LAN is enabled on a PC, the PC's network interface listens for a specially formed packet, even while the computer is powered off, hibernating, or in sleep mode. When the computer receives the packet, it powers on.
You could take advantage of Wake-on-LAN to turn on your PC over the Internet. This would allow you to leave your PC off except when you want to access its desktop or files remotely.
This can be done in two separate ways. You could forward the appropriate ports from your home router to the computer running on your local network, and then use a specialized software program to send the "Wake-on-LAN" packet to your home IP address.
Or, you could use a special software application that sits inside your local network. When it receives the signal, it sends the Wake-on-LAN packet to a computer on the same local network. TeamViewer (once again) offers this feature, allowing you to send Wake-on-LAN packets to other computers on your local network if one PC with TeamViewer enabled is powered on. TeamViewer has a complete guide to setting up Wake-on-LAN, from configuring the BIOS and network adapter to Windows and your home router. Much of the process is similar even if you're not using TeamViewer, although you'll need a mobile or desktop application that can send the Wake-on-LAN packets for you if you're opting not to use TeamViewer. The latest version of Parallels Access offers this feature, too.
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