Run Stored Procedure in SQL Developer?

With simple parameter types (i.e. not refcursors etc.) you can do something like this:

SET serveroutput on;
DECLARE
    InParam1 number;
    InParam2 number;
    OutParam1 varchar2(100);
    OutParam2 varchar2(100);
    OutParam3 varchar2(100);
    OutParam4 number;
BEGIN
    /* Assign values to IN parameters */
    InParam1 := 33;
    InParam2 := 89;

    /* Call procedure within package, identifying schema if necessary */
    schema.package.procedure(InParam1, InParam2,
        OutParam1, OutParam2, OutParam3, OutParam4);

    /* Display OUT parameters */
    dbms_output.put_line('OutParam1: ' || OutParam1);
    dbms_output.put_line('OutParam2: ' || OutParam2);
    dbms_output.put_line('OutParam3: ' || OutParam3);
    dbms_output.put_line('OutParam4: ' || OutParam4);
END;
/

---

Edited to use the OP's spec, and with an alternative approach to utilise :var bind variables:

var InParam1 number;
var InParam2 number;
var OutParam1 varchar2(100);
var OutParam2 varchar2(100);
var OutParam3 varchar2(100);
var OutParam4 number;

BEGIN
    /* Assign values to IN parameters */
    :InParam1 := 33;
    :InParam2 := 89;

    /* Call procedure within package, identifying schema if necessary */
    schema.package.procedure(:InParam1, :InParam2,
        :OutParam1, :OutParam2, :OutParam3, :OutParam4);
END;
/

-- Display OUT parameters
print :OutParam1;
print :OutParam2;
print :OutParam3;
print :OutParam4;

Executing easy. Getting the results can be hard.

Take a look at this question I asked Best way/tool to get the results from an oracle package procedure

The summary of it goes like this.

Assuming you had a Package named mypackage and procedure called getQuestions. It returns a refcursor and takes in string user name.

All you have to do is create new SQL File (file new). Set the connection and paste in the following and execute.

var r refcursor;
exec mypackage.getquestions(:r, 'OMG Ponies');
print r;