You can definitely host a Jira instance on DigitalOcean. Though Jira is commerical software, and you’ll need to purchase a license from them. They has an installation guide for Linux that you can find here.
They also provide an installer for the trial version that you can download onto your droplet and run. For instance, to download their 64bit trial edition, use:
Make the file executable and then run it with:
chmod +x atlassian-jira-6.2.4-x64.bin
When asked, select ‘Express Install’ After the installer has finished, make sure Jira is running by:
You should now be able to access Jira in the browser to finish installation at: