I had been using my own external hard disk for backing up my work computer, however (despite being 256GB) Time Machine kept complaining it didn't have enough space. Fortunately I also have access to a 3TB network share at work, so here's how I changed my Mac to backup to there instead.
Step 1: connect to the share
This is simple, either click 'Go' → 'Connect to Server…' or press ⌘(command)+K together.
smb://<server>/<share> into the box before connecting.
Step 2: Create a disk image on the network drive
Start by running disk utility.
Choose 'File' → 'Image' → 'Blank Image…' or press ⌘(command)+N together.
- Give it a sensible name (I chose "MacBook Pro Backup")
- Save it in the network share
- Give it a name (I chose "TimeMachine")
- A size (I went with 1TB)
- Make sure the Format is "Mac OS Extended (Journaled)"
- Choose encryption (and a passphrase) if you want
This may take a while for a large image.
Step 3: Make the remote share automatically mount
Add the network share and disk image to your account's login items:
'System Preferences' → 'Users & Groups' → 'Login Items'
Step 4: Make Time Machine use the disk image
In Terminal, run
sudo tmutil setdestination /Volumes/TimeMachine/ ("TimeMachine" is the name of the disk image when created in Disk Utility).
Time Machine should now be using the disk image for backups.