The content of a database is generally covered under copyright law, so broadly speaking… no. You cannot assume that copying and re-use is implicitly allowed by default.
Almost all major countries follow the Berne Convention. In the US (for example), almost everything published after April 1989 is considered "copyrighted" by default and protected whether it has a copyright notice or not. You should assume that any works that are not explicitly licensed for reuse may not be copied unless you know otherwise.
"Knowing otherwise" is where it gets tricky. There is a lot of data that simply is not copyrightable. But you have to understand copyright law and the laws governing reuse if you are going to act without explicit license. There are Fair Use provisions which allow a certain amount of re-use of original works. Copyright law doesn't generally protect mere listings of things (like ingredients, formulas, telephone listings, etc)… but copyright protection may extend to substantial literary expression within those listings. Some countries recognize separate property rights for databases which are somewhat distinct from copyright. Also, only original works of authorship are protected by copyright. Compilations of others' work may not.
If you don't see an explicit license, you should assume copyright by default — and then proceed cautiously to determine whether the work itself is actually copyrightable or if your application is covered under Fair Use.