If you can't upgrade to v10.7 Lion, you may be able to upgrade to v10.6 Snow Leopard which would give you access to the App Store.
The only way to purchase either Snow Leopard or Lion now is to contact Apple. Snow Leopard is no longer available from the Apple online store and Lion is not avaiable from the App Store.
Call 1-800-APL-CARE (1-800-275-2273)
You have to install Snow Leopard before you can install Lion or Mountain Lion.
Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
Whilst Apple have withdrawn Snow Leopard from their stores, you can still get it from Apple by calling 1-800-MY-APPLE (if you are in the USA) and they will supply the SL DVD for $20 for a single user, or $30 for a family pack that covers up to 5 Macs. You can also purchase the code to use to download Lion from the same number (Lion requires an Intel-based Mac with a Core 2 Duo, i3, i5, i7 or Xeon processor and 2GB of RAM, running the latest version of Snow Leopard), or you can purchase Mountain Lion from the App Store - if you can run that:
If you are outside the US call your national Apple Helpline:
If you're in the UK, use this number: 0871 508 4400
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard.
To use iCloud you have to upgrade all the way to Mountain Lion: