No such thing as a free lunch so wouldn't trust any free services.
I do sympatise with the customer of the OP but as well supported mail hosting can be had for as little as 12 quid a year plus maybe 2.99 for a domain I really do wonder why people still use @ISPname addresses if their intent is to shop around for the best deal.
That's why I registered my own domain and pay a small fee for mail hosting,. so am not tied to my broadband provider but in any case I do make sure I keep a copy of important mails on my PC.
Never ever rely 100% on a third party to store stuff long term that is important to you, hardware failures happen and backup procedures can go wrong.
I fear this kind of scenario will occur more and more as people put more and more trust in 'the cloud', trust which I personally feel is misguided.
Virgin (ADSL) => Namesco => Newnet => O2 => Plusnet => Zen => Newnet => Zen => Freeola => Vivaciti (using O2 Wholesale DSL) => Xilo (C&W Wholesale) => Xilo (O2 Wholesale)
Note: I don't lay turf for anyone. astro or otherwise, all views and opinions expressed are my own based on experience.