Not really, or at least i would be pretty afraid to use something like this in production. However, unserialize will use the autoload system or the function name specified in the unserialize_callback_func ini setting. So with a little hacking you can make this work:

// this a serialized object with the class "SomeMissingClass"
$str = 'O:16:"SomeMissingClass":1:{s:1:"a";s:1:"b";}';
ini_set('unserialize_callback_func', 'define_me'); // set your callback_function

// unserialize will pass in the desired class name
function define_me($classname) {
    // just create a class that has some nice accessors to it
    eval("class $classname extends ArrayObject {}");
}
$object = unserialize($str);
print $object['a']; // should print 'b'

You can use something like this to migrate your data to a little more handy format.

UPDATE:

I've consulted with my repressed memories on this (i've faced something like this once) and remembered an other solution:

So you have your SomeClass with a private property named $a

class SomeClass {
    private $a;
    public function getA(){
        return $this->a;
    }
}

And you have the serialized version of it:

$str = 'O:9:"SomeClass":1:{s:1:"a";s:1:"b";}';

When you unserialize it, and dump the result it will look like this, which is no good:

$a = unserialize($str);
var_dump($a->getA()); // prints 'null'
var_dump($a);
/* 
prints:
object(SomeClass)#1 (2) {
   ["a":"SomeClass":private]=>
   NULL
   ["a"]=>
   string(1) "b"
}
*/

Now, when an object get's unserialized, php will call it's __wakeup magic method. The data you need is there in the object, but not under the private property but a similarly named public one. You can't reach that with the $this->a since it will look for the wrong one, however the method get_object_vars() will return these properties and you can reassign them inside a __wakeup():

class SomeClass {
    private $a;
    public function getA(){
        return $this->a;
    }
    public function __wakeup(){
        foreach (get_object_vars($this) as $k => $v) {
            $this->{$k} = $v;
        }
    }
}
$str = 'O:9:"SomeClass":1:{s:1:"a";s:1:"b";}';
$a = unserialize($str);
print $a->getA();

I think it's needless to say that you should save yourself the further headache and convert your data to some dedicated data exchange format.