Verdict: Yes
Yes, but install LibreOffice instead. OpenOffice does have Linux packages, and it now gets very few updates. LibreOffice grew out of the same project, is far more active, and handles Microsoft Office files better.
Apache OpenOffice publishes Linux packages in both .deb and .rpm form, so the honest answer to whether it runs is yes. The reason to look elsewhere is the state of the project rather than the platform.
LibreOffice began as a branch of OpenOffice and took most of the developers with it. Since then LibreOffice has had a release every few months while OpenOffice has crept along, still on version 4.1.16. The practical result is that OpenOffice has fallen behind on the current Word and Excel formats.
LibreOffice is already installed on most beginner Linux systems, so switching usually means doing nothing. It opens every OpenOffice file you have, because both use the same OpenDocument formats, and it saves to .docx and .xlsx when you need to send something to an office.
Your .odt and .ods files come across untouched. Nothing is lost by leaving OpenOffice behind.
The same program grown up. Same menus, same file formats, and it is usually already installed. This is the one to use.
Your files: Opens every OpenOffice .odt, .ods and .odp file as is, and can also save .docx and .xlsx.
A ribbon layout closer to modern Microsoft Office. Worth it if Word files are what you handle all day.
Your files: Strongest with .docx and .xlsx. Opens OpenDocument files too.
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