My code isn't handling the exception (python)

Your try block doesn't catch an error because there is nothing wrong with assigning a variable a negative value. You need to check for this yourself and raise the appropriate error.

def __init__(self, width):
    if width < 0:
        raise ValueError('Width cannot be less than zero.')
    self.width = width

It's because negative width is a valid number and assign it to self.width does not raise ValueError. Instead of handing exception you can do a simple if check.

def __init__(self, width):
    if width < 0:
        print('Width cannot be less than zero.')
    else:
        self.width = width

you can use assert for raising error: assert width>0,'error'

You can try out this

class Square:    
    def __init__(self,width):
        try:
            if width < 0:
                 raise ValueError
            self.width = width
        except ValueError:
            print("Width cannot be less than zero."),
        print "Width is : %d" %width

    def __repr__(self):
        return 'Square({0})'.format(self.width)

    def __str__(self):
        return "Square(" + str(self.width) + ")"

obj = Square(10)  # pass positive value
obj = Square(-10) # pass negative value

Python does n't care whether width is zero or not. You have to take care of that.
You may rewrite your code in this way.

class Square:    

    def __init__(self,width):
        try:
            if width < 0:
                 raise ValueError("Negative value of width")
            self.width = width
        except ValueError:
            print("Width cannot be less than zero.")

    def __repr__(self):
        return 'Square({0})'.format(self.width)

    def __str__(self):
        return "Square(" + str(self.width) + ")"

What about this:

    self.width = width
    if self.width < 0:
        raise ValueError('Width cannot be less than zero.')