Is there any way to enable "feature access" when publishing a map using ArcGIS 10.1 server, without installing ArcSDE?
You don't need to install ArcSDE for enabling feature access capability of a map service in 10.1. You however would need to register a database as a geodatabase which will create multiple system tables (geodatabase schema) in this database. Starting with 10.1, you need to install ArcSDE only when you need to establish application server connection. If you are using direct connect, you are fine since all the libraries needed to access an enterprise geodatabase stored in a supported RDBMS are already shipped with ArcGIS Desktop and you don't need to install anything on top of that. More information on that here.
In 10.2, Esri has introduced a new feature. You can publish a map service with feature access based on a table with a spatial field (for instance, SQL Server native spatial type Geometry) and use this service for web editing without registering a database as geodatabase. More information on that here.