How do you define a class of constants in Java?

Use a final class, and define a private constructor to hide the public one.
For simplicity you may then use a static import to reuse your values in another class

public final class MyValues {

  private MyValues() {
    // No need to instantiate the class, we can hide its constructor
  }

  public static final String VALUE1 = "foo";
  public static final String VALUE2 = "bar";
}

in another class :

import static MyValues.*
//...

if (VALUE1.equals(variable)) {
  //...
}

Your clarification states: "I'm not going to use enums, I am not enumerating anything, just collecting some constants which are not related to each other in any way."

If the constants aren't related to each other at all, why do you want to collect them together? Put each constant in the class which it's most closely related to.