Hi!
I got the solution. Sadly I don't know the cause. This is going to be a lengthy explanation of what's going on, so please read through it all.
Serendipity can set specific read and write privileges for authorgroups, so that certain authors can only read what they have access to. By default serendipity ships with three authorgroups (Editor, Chief editor, Admin).
Users belong to usergroups. You can change group membership in the Group management panel, or you can also maintain your own group memberships in the "Personal Preferences" section. There, however, you can only withdraw from groups - you cannot simply join groups you are currently not a member of.
Now, those group memberships are important for two settings: The category section and the entry creation.
For each category, you can set which authorgroups are allowed to read a category or write to a category. By default that is set to the value "All Authors". So if you create a category "X" and assign read privileges to the Administrator group only, only he can read entries posted in that category in the frontend when he's logged in. For normal visitors, those entries will be HIDDEN. Also for members not of the group "Administrator", those postings will be HIDDEN.
You can enable or disable this permission management in the global Serendipity Configuration, the option is in "Appearance and Options" > "Apply read-permissions for categories".
Now there's a second option specific for entries that are NOT posted to categories. (i.e. where you did not select the category dropdown on the entry creation). For each entry, you can define which usergroups are allowed to read an entry. This is the same like with the category definition, but it can override settings individually. You can turn on/off this behavior via the configuration of the plugin "Extended options for entries" and then "Use group-based restrictions" as well as "Use user-based restrictions".
Now on to your specific blog: You seem to neither make use of categories (according to your database, none of the frontpage articles are assigned to any category), and you do not seem to make use of the feature to assign per-entry read access.
It might also be that you accidentaly deleted the category that all of your postings were associated to, thus leaving the permission system puzzled as to those postings that originally had reference to a group, and now no longer have.
Due to your old MySQL server version (3.23) it cannot properly handle database queries that return "NULL" values for unconfigured permissions.
So the serendipity database routine tries to tell the database "Give me all entries for the frontpage that might belong to a category; if it belongs to a category, check if the current authors has read privileges - if read privileges exist, nor no privileges are set at all, show that article". Now the last thing fails, because the old MySQL version is not able to differentiate a "NULL" value and an existing entryset.
I must admit I cannot tell you how it came to be that way, but it must have to do with category associations or usergroup memberships.
But there are some ways to fix this behaviour.
1. You could try to ask your webprovider kindly to upgrade to MySQL 4.0. MySQL 4.0 is stable since about 5 or 6 years, and MySQL 3.23 faded out of official support about 3 years ago. Just to put "your database server is old" into relation.

The upside of an upgrade would also be much faster performance.
2. You can DISABLE the option to apply read/write permissions on categories on global configuration, because you don't seem to use that feature
3. You can DISABLE the options in the plugin I mentioned (IF you have it installed!), because you don't seem to use that either.
Please try that out and tell me if it works for you!
Best regards,
Garvin