First of all, you need to comply with site standards. Many sites classify passwords for service accounts, etc, as Type 1 data (your Security team will tell you what Type 1 means). Breaches of handling policy for Type 1 data normally result in disiplinary action.

One method I have seen that complies with Type 1 handling policies is an encrypted Word document. Recent versions of Word support 128-bit encryption.

In Word, go to Tools -> Options. Click the Security tab, then the Advanced button. Select your desired encryption method (your site may have a mandate on what should be used), and set the key length to 128. This allows you to share the passphrase needed to open the document within the DBA team, and to change the passphrase at regular intervals. It can be cheaper and easier to user than some other methods.