Go to Administration > settings > pages and there choose "none" as basket page. There you go. Almost.
Next you have to delete variable $sMenu1 from file templates/container.tpl. This will disable whole upper-most menu and with it will go the last trace of basket link.
This should do it. Let me know it if worked as you wanted.
BTW, if you want to also totally disable prices, you should:
1. delete in file "templates/products_default.tpl" the <div id="price">...</div> (with all the content within)
2. delete in file "templates/pages_default.tpl" the <div class="price">...</div> (with all the content within)
If you want to have some products with price and some without, just create new templates from the pages_default.tpl and products_default.tpl templates and assign them to the products accordingly. I love QC 3.0, it's so versatile.
Will this work if you want to keep the basket but have no prices?
I.e. I want a catalogue for quotes, the client can browse and add items to a Quote basket and then at the end send us the quote order for us to get back to them with a final quote?
No it won't be work in this way. In Your case the simpliest way is to set prices to all of products to 0. You can make an if condition which will be checking is price equals 0. In case that this statment will be true You can replace displaying price by something. Another way is to replace in template place which displayng price by something or just clear this code (but it works only when all "products" are for free). You have to also make some changes in basket templates/orders_basket.tpl, templates/orders_form.tpl and templates/orders_print.tpl - erase places where prices are displaying. There should be simple to find.