Cloud storage services are everyday more used, they allows to easily and safely share own files over net, Dropbox, Amazon S3 and Box.com are some of them.

They have pros: can share files both privately and public with users, can sync them, they are available every moment; but also cons: you need time and use Internet for upload, the services have limitations (eg. max file sizes, max storage space).

If you simply need to access the files stored on your computer from Internet you can do it without using cloud within some free solutions.

TeamViewer
is a famous remote control software, it has also file transfer feature

Tonido
it allows to create your personal cloud. It’s multiplatform (Windows/Linux/Mac) and the interface is clean and nice. It allows to share files with other users, access via smartphone too, sync files, easy to setup (doesn’t requires any configuration), it’s secure (data are encrypted using SSL), can install apps that extend core features

Opera Unite
it’s a technology that turns the Opera browser into a personal web server. Over file sharing it also allows music streaming, show photo galleries and host web pages. You can read full guide here

DesktopNow (review/guide)
a new free sofware for Windows that allows to easily share a folder and access via web browser.
Once installed use the wizard that allows to create a free account first, then choose folder to share and after you can browse with your own local assigned IP address and specifying the port (by default is 140); the disavantage is that if you have a dynamic IP address you always must know it (or alternatively you can use a free service like No-IP to have an hostname that point to your IP address).

Gbridge
Windows software that allows to create a personal and encrypted VPN. Over secure file sharing there are also other features like desktop sharing, autosync, EasyBackup to easily setup backup auto recurring backup of your important folders. In order to work you need a Gmail account.