No, a company cannot suspend your GDPR rights – contracts can't override the law. Your rights as a data subject apply as long as your personal data is being processed.
However, there is no requirement in the GDPR that they fulfill your data subject rights through a self-service mechanism like a “download my data” button. They can require you to use another support channel. (But Google offers infamously bad support.)
In some cases, the service may legitimately decide that they cannot give you access to the data, for example if they believe that you are not the actual data subject (e.g. if they think that you hacked the account). The right to access must not adversely affect other people (Art 15(4)). If they have doubts about your identity, they can require further information to verify you (Art 12(6)). If your requests are excessive or unfounded (if you are spamming them), they can also turn down the requests (Art 12(5)).