Yes, it is possible.
I tried it a few years ago and came up with a simple, crude geocoder you can find on Github: https://github.com/plechi/mini-geocoder
You need a PostgreSQL-Server with installed PostGIS and HSTORE extensions.
Create a new Postgres Database.
Enable PostGIS and HSTORE extensions:
CREATE EXTENSION postgis;
CREATE EXTENSION hstore;
After that you need Osmosis and a dataset from Openstreetmap, look in the OSM-Wiki for details: http://wiki.openstreetmap.org/wiki/Downloading_data
Create the database schema from Osmosis (run commands in the terminal):
psql -d <database> -f <osmosis-folder>/script/pgsnapshot_schema_0.6.sql
Import the data (also execute in the terminal):
osmosis --read-xml file="<downloaded_datafile>.osm" --write-apidb host="<dbhost>" database="<dbname>" user="<dbuser>" password="<dbpassword>"
Theoretically, you could query the resulting database (examine the schema for details).
For my geocoder, I created a "optimization table" for easier queries:
CREATE TABLE geocode_optimized
AS SELECT
w.tags -> 'addr:street' AS street,
w.tags -> 'addr:housenumber' AS housenumber,
w.tags -> 'addr:postcode' AS postcode,
w.tags -> 'addr:city' AS city,
w.tags -> 'addr:country' AS country,
AVG(ST_X(n.geom)) AS longitude,
AVG(ST_Y(n.geom)) AS latitude,
to_tsvector(concat_ws(' ', w.tags -> 'addr:street',
w.tags -> 'addr:housenumber',
w.tags -> 'addr:postcode',
w.tags -> 'addr:city',
w.tags -> 'addr:country'
)) AS full_text,
st_makepoint(AVG(ST_X(n.geom)), AVG(ST_Y(n.geom))) AS point
FROM ways w
INNER JOIN way_nodes wn ON w.id = wn.way_id
INNER JOIN nodes n ON n.id = wn.node_id
WHERE exist(w.tags, 'addr:housenumber') AND exist(w.tags, 'addr:street')
GROUP BY housenumber, street, postcode, city, country;
CREATE INDEX idx_geocode_full_text ON geocode_optimized USING GIN (full_text);
Geocode (address to coordinates):
SELECT
street,
housenumber,
postcode,
city,
country,
longitude,
latitude
FROM geocode_optimized
WHERE full_text @@ plainto_tsquery('YOUR ADDRESS')
Reverse geocode (coordinates to address)
SELECT
street,
housenumber,
postcode,
city,
country,
longitude,
latitude,
CAST (st_distance_sphere(st_makepoint(longitude,latitude), st_makepoint('<longitude>','<latitude>')) AS FLOAT) as distance
FROM geocode_optimized;
As mentioned, this is quite crude and it is probably not the most performant solution.