Column: address   |  Type: Text
Column: geo_point | Type: geometry(Point,4326)

SELECT t.*, neighbor.*  -- or only selected columns
FROM   tbl t
LEFT   JOIN LATERAL (
   SELECT *
   FROM   tbl t1
   WHERE  ST_DWithin(t.geo_point, t1.geo_point, 5000 * 1.609344)
   ) neighbor ON true
WHERE t.address = 'my_address';  -- to restrict to a particular address

CREATE INDEX tbl_geo_point_gist ON tbl USING gist (geo_point);

SELECT t1.*, t2.*  -- or only selected columns
FROM   tbl t1
LEFT OUTER JOIN tbl AS t2
WHERE ST_DWithin(t1.geo_point, t2.geo_point, 5000 * 1.609344);