Sure it is possible to query multiple fields:
WHERE Facility1 = 'value1' AND Facility2 = 'value2' ...
That's standard SQL, you find it documented in the mysql manual as well:
For your specific problem (checkbox array values) it heavily depends. You need to map the checkbox array values to the fields in the mysql database to create an SQL statement that makes sense.
As you have not given any information next to that it's an array, there is not much to give as an example but that you map these array values to database columns.
$columns = array('Facility1' => 'checkboxname1', 'Facility2' => 'checkboxname2');
$query = 'SELECT * FROM rooms WHERE ';
$queryWhereAnd = array();
foreach($columns as $column => $name)
{
$queryWhereAnd[] = sprintf('%s = \'%s\'', $column, mysql_real_escape_string($_POST[$name]));
}
$query .= implode(' AND ', $queryWhereAnd ) . ';';