I was able to send a backup to my computer earlier this month on another computer. Oddly, I can't do now, however, as only a few fields in one row of a table have been affected since the backup, I have been able to restore the database on my local server the update the affected row manually, so that I now have a local clone of the database used for my web site. This means I can now run a local version of the site to test out changes before I make them live no the web site itself.
As the clone will reflect what is happening on the live site, I now no longer have a problem. Backups happen on the live site, but I am not able to access them myself - probably because the data base is a shared one.
I have no idea what has caused the problem - I did change outbound firewall rules on the router for port 1433 (disabling it for outbound) but I don't think that should affect matters.
I rashly accepted the post of webmaster at the AGM of a model aircraft association at is AGM (no one else would volunteer)- and found somewhat to my surprise a website, with no CMS, and based on asp.net. I am retired - and it is a few years since I coded a couple of asp.net sites. Fortunately, this one is back in my era - uses asp.net 3.5, and it is information only, with no user input. I may change that - but if I do I will make sure I have better control over database backup. Talking of which, the site one is used to supply web page content, but it is not relational; all the tables are unrelated tables - flat files would have worked just as well. For the meantime, I can now see that I have no immediate need to transfer further backups of the database to my computer. When I do, I shall change web packages/hosts to make sure that the facility is readily available.
In the long term, I will probably switch the site to a CMS, and encourage user input - but in the meantime I have quite a lot of tidying up to do.
Thanks djj55 for the helpful discussion I received on this thread, it was appreciated.