Handling a class with a long initialization list and multiple constructors?

How about refactor the common fields into a base class. The default constructor for the base class would handle initialization for the plethora of default fields. Would look something like this:

class BaseClass {
    public:
    BaseClass();
};

class Object : public BaseClass
{
    Object();
    Object(const string &Name);
    Object (const string &Name, const string &path);
    Object (const string &Name, const bool loadMetadata);
    Object (const string &Name, const string &path, const bool loadMetadata);
};

BaseClass::BaseClass() :
    parent_index (0),
    rowData (new MemoryRow()),
    objectFile (),
    rows (new MemoryColumn (object_constants::RowName, OBJECTID, object_constants::ROWS_OID)),
    cols (new MemoryColumn (object_constants::ColName, OBJECTID, object_constants::COLS_OID)),
    objectName (new MemoryColumn(object_constants::ObjName, STRING, object_constants::short_name_len, object_constants::OBJECTNAME_OID)),
    parent     (new MemoryColumn(object_constants::ParentName, STRING, object_constants::long_name_len, object_constants::PARENT_OID)),
    parentIndex (new MemoryColumn(object_constants::ParentIndex, OBJECTID, object_constants::PARENTINDEX_OID)),
    childCount (new MemoryColumn (object_constants::ChildCount, INTEGER, object_constants::CHILD_COUNT_OID)),
    childList (new MemoryColumn (object_constants::ChildList, STRING, object_constants::long_name_len, object_constants::CHILD_OID)),
    columnNames (new MemoryColumn (object_constants::ColumnNames, STRING, object_constats::short_name_len, object_constants::COLUMN_NAME)),
    columnTypes (new MemoryColumn (object_constants::ColumnTypes, INTEGER, object_constants::COLUMN_TYPE)),
    columnSizes (new MemoryColumn (object_constants::ColumnSizes, INTEGER, object_constants::COLUMN_SIZE))
{}

Your Object constructors should look a little more manageable, now:

Object::Object() : BaseClass() {}
Object::Object (const string &Name): BaseClass(), name(Name) {}
Object::Object (const string &Name, const string &path): BaseClass(), name(Name), path_(path){}
Object::Object (const string &Name, const bool loadMetadata): BaseClass(), name(Name){}
Object::Object (const string &Name, const string &path, const bool loadMetadata): BaseClass(), path_(path) {}

Similar in nature to Iraimbilanja's answer, but avoids adding an inner-class for accessing data, which might impact a lot of existing code. If you've already got a class hierarchy, though, it may be difficult to factor it into a base class.

Yes, it's possible.
For simplicity I'll pretend that the original code is:

class Foo {
public:
    Foo() : a(0), b(1), x() { }
    Foo(int x) : a(0), b(1), x(x) { }

    int get_a() const { return a; }
    int get_b() const { return b; }
    int get_x() const { return x; }
private:
    int a, b, x;
};

The refactored code, then, is:

class Foo {
public:
    Foo() : x() { }
    Foo(int x) : x(x) { }

    int get_a() const { return common.a; }
    int get_b() const { return common.b; }
    int get_x() const { return x; }
private:
    struct Common {
        Common() : a(0), b(1) { }
        int a, b;
    } common;
    int x;
};

Now a couple of years later we have C++ 11. If you can use it in your project you have two options:

When the common initialization values are only known at runtime you can use delegating constructors, which means one constructor calls another.

 // function that gives us the init value at runtime.
 int getInitValue();

 class Foo
 {
     const int constant;
     int userSet;

 public:
     // initialize long member list with runtime values
     Foo() 
       : constant(getInitValue())
       , userSet(getInitValue())
     {}

     // other constructors with arguments
     Foo( int userSetArg) 
       : Foo()
       , userSet(userSetArg) 
     {
     }
 };

or you can initialize the members directly in the class definition if their values are known at compile time.

class Foo
{
    const int constant = 0;
    int userSet = 0;

public:
    Foo( int userSetArg) : userSet(userSetArg){}
}