You only need Windows SharePoint Services (WSS) to do that. SharePoint
Portal Server is intended to aggregate and organize information for an
enterprise. WSS is intended for people  to collaborate on documents (as
you describe).

I thought the question was more "can you keep a document  private to 2 or 3
people" and the answer is yes and no

You cannot secure at an individual document level in sharepoint  2003 (and hence
no) but if you place the document in its own document library then you can
restrict access to that document library (and thus the document) to only 2 or 3
people (hence yes)

Hope that's what you were after.

So from your answer I
understand that inside SPS if I create my own document  library and
put a document inside it and add users that I want to share the
document, and then uncheck the anonymous access and request access to
the document library it should be accessible only to the users that I
added, right. I tried this but still it is viewable by all the
anonymous users whom I gave this link to in our intranet. Please help
me in finding out how can I achieve doc sharing only to a limited
users.

You cannot set individual permissions on document  libraries in SPS. You
must use a WSS document library.

I tried achieving document  sharing to limited audience by
following the following steps but was unsuccessful:

Set up for properties for the document library/list enabled/disabled:

Inside "Modify settings and columns" select "Change permissions for
the list", here you can add all the individual users you want to
share the document with.

1. Change anonymous access - All checkboxes unmarked
2. Inside Modify item-level security, under Item-level Permissions
please select
Read-access- Only their own
Edit-access- None or Only their own

This is in response to "You cannot secure at an individual document
level in sharepoint  2003 but if you place the document in its own
document library or a list then you can restrict access to that
document library/list (and thus
the document) to only 2 or 3 people."

You cannot set individual permissions on document  libraries in SPS. You must use
a WSS document library.