You need to extract the number from the string, and pass it into the Date constructor
:
var x = [{
"id": 1,
"start": "\/Date(1238540400000)\/"
}, {
"id": 2,
"start": "\/Date(1238626800000)\/"
}];
var myDate = new Date(x[0].start.match(/\d+/)[0] * 1);
The parts are:
x[0].start - get the string from the JSON
x[0].start.match(/\d+/)[0] - extract the numeric part
x[0].start.match(/\d+/)[0] * 1 - convert it to a numeric type
new Date(x[0].start.match(/\d+/)[0] * 1)) - Create a date object