Static DataTable or DataSet in a class - bad idea?

Question

I have several instances of a class. Each instance stores data in a common database. So, I thought "I'll make the DataTable table field static, that way every instance can just add/modify rows to its own table field, but all the data will actually be in one place!"

However, apparently it's a bad idea to do use static fields, especially if it's databases: Don't Use "Static" in C#?

Is this a bad idea? Will I run into problems later on if I use it?

This is a small project so I can accept no testing as a compromise if that is the only drawback. The benefit of using a static database is that there can be many objects of type MyClass, but only one table they all talk to, so a static field seems to be an implementation of exactly this, while keeping syntax concise.

I don't see why I shouldn't use a static field (although I wouldn't really know) but if I had to, the best alternative I can think of is creating one DataTable, and passing a reference to it when creating each instance of MyClass, perhaps as a constructor parameter.

But is this really an improvement? It seems less intuitive than a static field.