I think you'll find that this issue isn't unique to Chrome. We have a fault ticket raised for intermittent webmail access issues, ref: F004841712. We are working on getting this resolved but it might be worth you considering setting up a client for use in the meantime.
Please take a look at our help pages on how to set up an email client and let me know if you need any help with that. Or post a nudge on here if you want an update on the webmail fault.