Calculate total area of polygons in shapefile using GDAL?

There's a special field in OGR SQL called OGR_GEOM_AREA which returns the area of the feature's geometry:

ogrinfo -sql "SELECT SUM(OGR_GEOM_AREA) AS TOTAL_AREA FROM myshapefile" myshapefile.shp

where TOTAL_AREA unit of measure depends by the layer SRS (read the comments below).

Yes, it is possible, but you need to use the OGR SQLite dialect as follows:

ogrinfo -dialect SQLite -sql 'SELECT SUM(ST_Area(geometry))/10000 FROM myshapefile' myshapefile.shp

Also, ensure that myshapefile is the layer name in myshapefile.shp. You can do this as follows:

ogrinfo myshapefile.shp

INFO: Open of `myshapefile.shp`
    using driver `ESRI Shapefile' successful.
1: myshapefile (Polygon)

Using QGIS, you may run this simple code for printing the total area of the shapefile (I assume you are evaluating the area in a projected reference system):

from qgis.core import *

filepath = 'C:/Users/path_to_the_shapefile/shapefile.shp'
layer = QgsVectorLayer(filepath, 'layer' , 'ogr')

area = 0
for feat in layer.getFeatures():
    area += feat.geometry().area()

print area # total area in square meters

print area/10000 # total area in hectares