How do I create a static local variable in Java?

You can have a static class variable, which will be preserved across all instances of the class. If that's what you want. If not, use an instance variable, which will only be preserved across method calls on this object.

public class Foo {
   static int bar;
   //set bar somewhere

   public int baz() {
      return 3 * bar;
   }
} 

If you want to reuse variable value between function calls and isolate this variable from other methods, you should encapsulate it in an object. First, define a class with only the "static-like" variables and the function you need:

class MyFunctionWithState {
    private int myVar = 0;
    public int call() {
      myVar++;
      return myVar;
    }
 }

Then, from your client code:

class Foo {
    private MyFunctionWithState func = new MyFunctionWithState();
    public int bar() {
      return func.call();
    }
 }

Now if func relies on the internal state of Foo you can either pass the relevant data through call() or pass an instance of Foo and let the function call the appropriate getters:

class MyFunctionWithState {
    private int myVar = 0;
    public int call( Foo f ) {
      myVar += f.getBaz();
      return myVar;
    }
 }

class Foo {
    private MyFunctionWithState func = new MyFunctionWithState();
    public int bar() {
      return func.call( this );
    }
    public int getBaz() {  /*...*/  }
 }

Should I resort to using instance variables?

yes - after all, that is what instance variables are for. They store the state between invocations of the methods of the object. Having static class variables can sort of achieve the same result, but can make your program more coupled and harder to test/mock/maintain.

Disclaimer: Just because you can, doesn't mean you should

Although Java doesn't support static local variables, it's fairly easy to emulate in recent versions of Java. From Java 16 onwards, it is possible to create a local class with a mutable static field in a method. Subsequently, you mutate the local class's static field just like you would in C/C++.

public class Main {

    public void foo() {
        class Local {
            static int count = 0;
        }
        Local.count++;
    }
    
}

Before Java 16, local classes may only contain static constants.