Sears, Roebuck and Co., the nation's largest retailer, said yesterday its earnings fell 36 percent to $135.5 million in the third quarter from $212.1 million in the same period last year.

The company said earnings per share for the quarter ended Oct. 31 were 43 cents on revenues of $6.46 billion. That compares with per-share earnings of 67 cents and revenues of $6.33 billion in the year-earlier quarter. e

Net income for the nine months ended Oct. 31 was $317.4 million ($1.04 a share) compared with $555.7 million ($1.74) last year, and revenues were $17.93 billion compared with $17.61 billion.

Federated Department Stores Inc. earned $2.11 a share in its third quarter ended Nov. 1, up from $1 a year earlier, on a rise in sales to $1.534 billion from $1.415 billion. Net income rose to $105.45 million from $48.19 million.

Nine-month profits were $155.83 million ($3.22 a share) on sales of $4.202 billion compared with $98.88 million ($2.05) a year earlier on sales of $3.888 billion.

All of the third-quarter profit gain was on the sale of equities in shopping centers. Third-quarter operating profit actually was down 1.7 percent from a year earlier.

May Department Stores Co., which owns the Hecht Co. of Washington and Baltimore, yesterday reported that for the first time in its history its nine-month sales surpassed $2 billion. Sales for the third quarter were $775 million, bringing the year-to-date total to $2.1 billion. Sales for the third quarter of 1979 were $718.6 million.

Net income for the third quarter was $23.1 million (79 cents a share) compared with $24.1 million (83 cents) in the 1979 third quarter.

Nine-month sales rose 6.6 percent to $2.1 billion, while earnings per share fell 5.9 percent to $1.74 from $1.85.

Carter Hawley Hale Stores Inc. reported yesterday an increase in third-quarter sales of 8 percent to $630.3 million from $583.4 million last year. Net earnings fell 13 percent to $10.5 million (37 cents a share).

Nine-month net earnings were $25 million compared with $26.8 million reported for the 1979 period, a decline of 6.5 percent. Earnings per share for the nine-month period were 87 cents compared with 97 cents in 1979.