After a short familiarisation time, Lineolated Parakeets soon get confiding and driven by their immense curiosity they are inspecting their surrounding. This exploratory behaviour, you should comply with a sufficiently large accommodation and/or free flight in your apartment.
But what means "sufficiently large"?

In the German
animal welfare act (Confederacy, i. t. e. of the bulletin of May 25, 1998) it is written:
"Animal keeping
The one who keeps an animal, is looking after it or has to look after it,
  1. must feed, look after it and accommodate the animal appropriate to its species and its needs,
  2. must not restrict the animal's possibility to move naturally, so that it has pain or avoidable suffering or injuries,
  3. must have the appropriate knowledge and ability to feed, look after and accommodate the animal due to its natural needs."

With the above-stated, the essential is said. But there is the problem, that every single keeper of parakeets or other animals considers "appropriate" and "natural-like" very subjective and that he defines it according to his thoughts.

Therefore, the ministry for food, agriculture and forestry passed on January 10th, 1995 the "
Minimum requirements for the care of parrots" (English translation of the German original text!), which is founded on a advisory opinion about the appropriate keeping of birds. This serves keepers of parrots to examine their own keeping conditions and therefore, it refers to that §2 of the animal welfare act. Inter alia there is stated:
"Apart from exceptions parrots are living in pairs or in groups. Basically they have to be kept like that in the charge of the human being. Excluded from this are quarrelsome and at present existing, completely adjusted on man as well as ill or injured birds. In the future, when selling parrots it must be referred to the necessary couple keeping and therefore, as a rule they must be only handed over in pairs. Young birds should be raised, so that they are species-fixed." (I. General Part, section 5)

Particularly for parakeets with a total length up to 25 cm, the following measures for cages and aviaries are considered as minimum:
1.0 x 0.5 x 0.5 m (length x width x height)
"The stated measures for cages and aviaries count for the placement in pairs and must not be shorten even at explained single keeping. The base (floor space) have to be extended by 50 % for every additional pair. Cages must be placed at a minimum height of 80 cm." (II. Particular Part, A. General keeping demands, section 2)

Well, as you can see, everything is clearly defined.
Though, in my opinion, it does not clearly result from this recommendation, if these measures refer to a keeping with free flight or a steady keeping in an aviary. Possibly it is not distinguished between these two keeping possibilities. However, then I personally consider these recommended measures as unacceptable, because without free flight a floor space of half a square metre is definitely not enough, nor for a Lineolated Parakeet.
Many breeders consider 1 to 1.5 square metres floor space per couple as absolute minimum for a long-term keeping of the Lineolated Parakeet.

Arndt (1986) wrote about it: "The accommodation seems to be of greater importance for most of the thick-billed parakeets than for many other parrot species. Their small height tempt many breeders to accommodate them, similar to the Agapornis-species or the parrotlets, in small aviaries or box cages. With that you do not deal with their lively nature which they only show in a spacious flight aviary. (...) Among the B. lineola and B. aymara, the chances of hatching success are distinctively higher in a spacious aviary.
Outdoor aviaries with a shelter are perfect. Per couple the minimum size should not remain under 2.5 square metres, and you should add 1 square metre for every couple in a communal keeping."

And further:
"Most breeders keep their parakeets in an aviary, which complies with the nature of the small Lineolated Parakeet. They are considerably more lively. For keeping in box cages, these should not be smaller than 1.2 m x 0.6 m x 0.6 m. "