Hi Lukas,
Thank you for posting your query in Microsoft Office Community.
A quick question before we proceed.
-Which browser are you using?
You may go to Redeem your product key for renewal from My Account page as
described in article below and check if it helps.
https://support.office.com/en-us/article/Renew-Office-365-for-home-58f3d0f8-15c4-4ad8-8c5f-16d682e8b545
If you're already using Internet Explorer, try the following and verify the result:
Start an InPrivate browser session and then try downloading installing Office 365 and verify the result.
InPrivate Browsing helps prevent your browsing history, temporary Internet files, form data, cookies, and user names and passwords from being retained by the browser. Internet Explorer will launch a new browser session that won’t
keep any information about webpages you visit or searches you perform. Closing the browser window will end your
InPrivate Browsing session. To start an InPrivate Browsing session:
a) Go to the Tools (Press Alt + T) menu and select InPrivate Browsing.
b) Use the InPrivate Browsing window to redeem key as mentioned in article below.
c) Make sure that you're signed out from any Microsoft services.
If issue still persists, reset Internet Explorer's setting following the steps mentioned in the following article then try to install Office and verify the result:
http://support.microsoft.com/kb/923737
Disclaimer: The Reset Internet Explorer Settings feature might reset security settings or privacy settings that you added to the list of Trusted Sites. The Reset Internet Explorer Settings feature might also reset parental
control settings. We recommend that you note these sites before you use the Reset Internet Explorer Settings feature.
If you have not tried downloading the Office suite through a different web browser, I suggest you to try the same and see if it helps.
I hope the above information helps. Try the suggestions and let me know the result at your earliest convenience.
Thank you.