This will depend on what you know about the String values you will be working with. For example...

// if the year is always represented by the left-most four characters, then you can use the left() method...
String rawValue = '199924';
String yearValue = rawValue.left(4);
system.assertEquals('1999', yearValue);

// week number can be obtained by using the removeStart() string method after obtaining the year... String String weekValue = rawValue.removeStart(yearValue);
system.assertEquals('24', weekValue);

// the left method will work in this instance, even if the week number is sometimes represented by a single digit...
rawValue = '20014';
yearValue = rawValue.left(4);
weekValue = rawValue.removeStart(yearValue); 
system.assertEquals('2001', yearValue);
system.assertEquals('4', weekValue);

// my approach from here would be to find the start of the year, and perhaps that start of first week of the year, assuming the week that contains January 1st is week #1
Integer yearInt = Integer.valueOf(yearValue);
Integer weekNumber = Integer.valueOf(weekValue);

// build a Date instance of January 1 of that year, then of the start of that week...
Date startOfYear = Date.newInstance(yearInt, 1, 1);
Date startOfWeekOne = startOfYear.toStartOfWeek();
Date startOfWeekFour = startOfWeekOne.addDays(weekNumber * 7);

system.assertEquals(Date.newInstance(2001, 1, 28), startOfWeekFour);

If the year is ever represented as two characters, then you would need some way of determining how many of the starting characters represent the year value.

Please double-check my assuption regarding whether the start of week one is the start of the week that contains January 1st.

My recommendation would be to create a utility method that can return the date for you, when passing in the raw string value, then fully test this in a test method, passing in valid and invalid values. You'll need to determine what the response should be whenever the inputs are invalid.